- Non-degenerate Hermitian Form

: Analysis of forms and properties of inner product spaces in complex and real vector spaces

Complex-valued non-degenerate Hermitian form $(\cdot \mid \cdot)$ or

Real-valued non-degenerate symmetric form $(\cdot \mid \cdot)$

1. Linearity in the Second Argument #

$$(u \mid v_1 + c\ v_2) \ =\ (u \mid v_1) \ +\ c\ (u \mid v_2)$$

2. Hermiticity #

Complex vector space: Hermiticity, i.e.,

$$(u \mid v) \ =\ (v \mid u)^*$$

Or real vector space: symmetry, i.e.,

$$(u \mid v) \ =\ (v \mid u)$$

Note that from 1 and 2, the first argument of $(\cdot \mid \cdot)$ is automatically conjugate-linear.

If it’s a real vector space, it’s linear, meaning $(\cdot \mid \cdot)$ becomes a bilinear form.

3. Non-degeneracy #

Non-degeneracy: For all vectors $v$ in the vector space, $(w \mid v)$ equals 0 only when $w=0$, which means for all $v$,

$$(u + w \mid v) \ =\ (u \mid v)$$

is satisfied only when the vector $w$ is the zero vector. If there exists a non-zero vector $w$ satisfying this condition, $(\cdot \mid \cdot)$ is said to be degenerate. In other words, any two different vectors $v_1, v_2$ in the vector space are always distinguishable by $(\cdot \mid \cdot)$. That is, for any vector $w$ in the vector space,

$$(v_1 \mid w) \ =\ (v_2 \mid w)$$

there do not exist two different vectors $v_1, v_2$ for which this always holds.

In a real vector space, if conditions 1, 2, and 3 are satisfied, it becomes a metric space.

That is, Euclidean space and Minkowski space are metric spaces, but complex space is not a metric space.

cf.) Mathematically, if we define $$d(u, v) \ =\ ||\ u - v\ ||\ =\ \sqrt{(u - v \mid u - v)}$$ then d becomes a distance function, and complex inner product space is also a metric space with the distance function d.

4. Positive-definiteness #

For all non-zero vectors $v$ in the vector space,

$$(v \mid v) \ >\ 0$$

To be an inner product space, condition 4 must also be satisfied.

5. Additional Notes #

Therefore, a real vector space that satisfies 1, 2, 3 but not 4 is a Minkowski (space)time (metric space, not an inner product space), a real vector space that satisfies up to 4 is a Euclidean inner product space (metric space), and a complex vector space that satisfies all of 1, 2, 3, 4 is a complex inner product space (Hilbert space, not a metric space). If $(\cdot \mid \cdot)$ satisfies all of 1, 2, 3, 4, it represents an inner product, and if it’s defined in a real vector space satisfying 1, 2, 3, it’s a metric.

Special relativity theory represents various observables as 4-vectors in Minkowski spacetime. The state vectors of quantum mechanics exist in a complex inner product space (Hilbert space). Quantum field theory, which combines special relativity, deals with observable vectors in Minkowski spacetime and state vectors in Hilbert space. This is exciting.